Page Rank and Pagerank(tm) - Toolbar PR and Actual PR

Introduction
I realize that there is quite a bit of confusion when it comes to the terms Page Rank and Pagerank™. People often refer to the terms in a way that is ambiguous. The problem is that the meanings of the two terms are very different.

The best way to avoid confusion is to expand the term "Page Rank" when you are using it in a conversation or an article in order to clarify your meaning. It is best to use the terms "Google Pagerank™ and "Page Ranking" or "Ranking of a Page" for clarification.

But what exactly are the meanings of the terms, and why is one of the two inaccurate or incorrect most of the time when displayed via a certain very popular browser plug-in or specialized SEO tools?

"Page Rank" (two words), incorporates ALL ranking criteria used by a search engine algorithm to calculate the value a page has for a specific search term. All pages that match a key phrase are sorted by the page rank value.
This is part of every search engine that displays search results by "relevance" and not just alphabetically.

"Pagerank" (PR) - the Google Pagerank™, is a numeric value that represents how important a page is on the web. Pagerank™ is a trademark term of Google and the algorithm named Pagerank™ is patented by the Google founders Sergey Brin and Lawrence Page. Google figures that when one page links to another page, it is effectively casting a vote for the other page. The famous Pagerank™ algorithm allows the determination of each page's Pagerank™ in relationship to all pages available on the web (in the Google Index).

The Google Pagerank™ is only one of over 100 criteria for Google to determine the "Page Rank" of a page for a specific key phrase in Google's vast search index.

Google Toolbar PR vs. Actual PR
The Google Pagerank™ value displayed in the Google Toolbar (the value between 0-10) is not the value used by Google internally. That means, for example, that a page where the toolbar shows "0" as Pagerank™ might have internally a much higher Pagerank™, between 3 and 4 for example. The reason for the difference is not clear and people can only speculate.

Even if the displayed Pagerank™ value is somewhat accurate, it is still not the exact number/value that Google is using in its algorithm.

The actual calculated number using the original algorithm is a floating point number with a lot of digits after the decimal point. A site with PR 1 could have an actual PR of 1.298245, while another site with PR1 might only have an actual PR of 1.00123.

Integer vs. Floating Point
The fact that the toolbar value is rounded to the next whole number is important, because this invisible difference is making a bigger difference the higher the Pagerank™ of a site becomes (a difference of hundreds of thousands of inbound links between two pages with the same displayed PR).

Furthermore, the "Google Pagerank™" is a number/value assigned by Google to a page/document, independent from the keyword phrase. "Page Rank" on the other hand is the number/value assigned to a page in combination with a specific keyword phrase.

Conclusions

Fact 1.
"Page Rank" is not the same as "Google Pagerank™."

Fact 2.
"Google Pagerank™ shown in the Google Toolbar is not the same as the actual "Google Pagerank™" number or value used by Google to determine the "Page Rank" of a page for a specific key phrase.

Fact 3.
"Google Pagerank™" value is for a specific page or document on the web and the "Page Rank" value is for a specific web page/document and keyword phrase combination.

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